Essien was snapped up by Madrid on a season-long loan from Chelsea late on transfer deadline day, with the Spanish giants also agreeing on Friday to let Lassana Diarra join Russian side Anzhi Makhachkala for the current campaign.
The 29-year-old Essien's move to the Bernabeu sees him rejoin former Blues boss Mourinho, who brought him to England in 2005, and the Ghana international is delighted to be linking back up with someone he considers a father figure.
"I call him that (daddy) because I see him as like a father. He took me to Chelsea and we have a very good relationship. I understand him very well," said Essien, who passed his medical in Madrid on Saturday before penning his contract at the Bernabeu.
"I was in my hotel room yesterday [Friday] and I had a call from my 'daddy' and he said he would love to have me here.
"It's been all very quick but I'm very happy."
The versatile midfielder has made 247 appearances for Chelsea, scoring 25 goals and twice winning the club's goal of the season award.
He also helped the London outfit to two Premier League titles, four FA Cups and last season's Champions League title.
Essien has suffered with injuries in recent years but insists he is fit and ready to go with Madrid, saying: "I've had a very good pre-season with Chelsea and I'm ready to play.
"I've had my injuries but they are part of the past. I'm fit and strong. If not, Real Madrid would not have signed me.
"If they want me to play, I will play tomorrow [Saturday].
"Daddy knows me very well, he knows I can play in any position he wants me to, except goalkeeper."
Regarding his future at the end of the campaign, Essien said: "At the moment I'm only on loan and I'm going to work hard for Real Madrid. At the end of the season we'll see what happens."
Essien has been included in Madrid's squad for the visit of Granada tomorrow, where the defending champions will be looking to pick up their first league victory after drawing their opening match at home to Valencia before losing at Getafe last weekend.
Madrid, who on Monday signed Luka Modric from Tottenham, go into the match boosted by their 2-1 home victory over Barcelona on Wednesday which saw them win the Spanish Supercopa on away goals.